Analysis of heave motions of a truss spar platform semi-closed moon pool

Ocean Engineering 92 (2014) 162-174

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Analysis of heave motions of a truss spar platform with

semi-closed moon pool

Liqin Liu^'""'* Atilla Incecik'', Yongheng Zhang^ Yougang Tang"*

' state Key Laboratory of Hydraulic Engineering Simulation and Safety Tianjin University, Tianjin 300072, Cliina ^ Department of Naval Architecture, Ocean and Marine Engineering, University of Strathclyde, Glasgow G4 OLZ, UK

®

A R T I C L E I N F O

Article history:

Received 29 April 2014 Accepted 7 September 2014 Available online 23 October 2014

Keywords:

Truss spar platform Semi-closed moon pool Heave motions Dynamic coupling Model experiments

A B S T R A C T

T h i s p a p e r p r e s e n t s the h e a v e m o t i o n of a truss Spar p l a t f o r m in r e g u l a r w a v e s c o n s i d e r i n g the d y n a m i c c o u p l i n g b e t w e e n the m o t i o n s of the p l a t f o r m a n d w a t e r i n the m o o n pool. For a s p a r platform w i t h s e m i - c l o s e d m o o n pool, w a t e r c a n f l o w freely t h r o u g h the guide plate at bottom of t h e m o o n pool. T h e m a s s of w a t e r inside t h e m o o n pool is c o m p a r a b l e to the p l a t f o r m m a s s itself if the top tension riser s y s t e m is used. T h e n t h e effect of the m o o n pool w a t e r o n the m o t i o n s of the p l a t f o r m s h o u l d not be ignored. In the s t u d y p r e s e n t e d in the paper, a 2 - D O F (Degree of F r e e d o m ) d y n a m i c c o u p l i n g equations of the h e a v e motions of the s p a r p l a t f o r m a n d the v e r t i c a l motions of w a t e r in the m o o n pool w e r e d e r i v e d c o n s i d e r i n g t h a t the m o o n pool is totally closed, or 30% or 70% open. T h e results s h o w that motions of w a t e r in the m o o n pool significantiy affect the h e a v e motions of t h e s p a r p l a t f o r m . P a r a m e t r i c s t u d y w a s also c a r r i e d out to find the effect of the ratio o f o p e n i n g of t h e b o t t o m plate o n the c o u p l e d motions. F i n a l l y t h e m o d e l e x p e r i m e n t s w e r e c a r r i e d out to v a l i d a t e the n u m e r i c a l c a l c u l a t i o n s .

© 2 0 1 4 E l s e v i e r Ltd. A l l rights r e s e r v e d .

1. Introduction

Spar platforms are one of the platform types used to produce oil and gas f r o m deep-sea fields. Research investigations on Spar platforms mainly focus on wave loading on the hull and its hydrodynamic characteristics, the dynamic response of the plat-form, stability analysis of coupled heave and pitch motions, simulation of the mooring and the riser system and the vortex-induced vibration (Tang, 2008). Investigators so far have carried out extensive research on the coupled motions between the platform hull and its mooring and riser systems, as well as on the coupling effects of different degrees of freedoms (Kim et al„ 2001; Tao et al., 2004; Mohammed et al., 2011; Xu and Jing, 2013). On the other hand the investigations to study the coupled motions between the platform and water inside the moon pool are rare.

Risers and drilling string go through a moon pool which is located i n the centre of the hull of a spar platform and extends f r o m the bottom of the hull to the deck. A guide plate is placed at the bottom of the moon pool (Drobyshevski, 2004; Kristiansen and Faltinsen, 2012). According to the design requirements, the moon

* Corresponding author at: State Key Laboratory of Hydraulic Engineering Simulation and Safety Tianjin University, Tianjin 300072, China.

T e L : + 8 6 22 27406074 811.

E-mail addresses: liuJiqin@eyou.com, llqin.liu@strath.ac.uk (L. Liu).

http://dx.doi.Org/10.1016/j.oceaneng.2014.09.048 0 0 2 9 - 8 0 1 8 / © 2014 Elsevier Ltd. All rights reserved.

pool is sometimes designed to be semi-closed, such as Holstein Spar platform, which allows water to flow freely i n and out of the moon pool. There are two common ways to deal w i t h the water inside a semi-closed moon pool (Gupta et al., 2008): (1) the moon pool is completely closed and the water inside the moon pool is trapped and thus it moves as a rigid body together w i t h the platform; (2) the moon pool is completely open to the sea and the presence of the water inside the moon pool w i l l not significantly affect the heave motions of the platform. The water in the moon pool has two kinds of natural vibration modes, which are piston-like motion i n vertical direction and sloshing motion caused by the free surface (Faltinsen and Timokha, 2009). If the top tension risers are used, the mass of the water inside moon pool is comparable to the mass of the platform. The effect of the motions of water inside the moon pool on the motions of the Spar platform should not be ignored.

Aalbers (1984) assumed that the water column inside a floating vessel can be replaced by a frictionless piston, then the motion equation of water inside the moon pool was derived and the coupled motions were studied. It was found that the water inside the moon pool can decrease the heave amplitude of the vessel. Barreira et al. (2005) treated the motions of water inside a moon pool as a spring-mass system and studied the coupled motions of a mono-column structure and water inside a moon pool. They found that the heave motions of the platform can be decreased by an

L Liu et al. / Ocean Engineering 92 (2014) 162-174 163

appropriate design of a guide plate. Sphaier et al. (2007) presented the mono-column dynamic behavior in waves as obtained f r o m model experiments. They measured the mono-column vertical motions and the motions of water i n the moon pool for different opening ratios of the guide plate. The effect of guide plate opening ratios on the heave motions of the platform was analyzed. Gupta et al. (2008) derived a 2-DOF coupling dynamic model of a spar heave and the vertical motion of water inside the moon pool. This work highlights the importance of the moon pool hydrodynamic characteristics i n predicting the heave motions of a spar platform. Spanos et al. (2011) developed a 6-DOF nonlinear model, including surge, heave, and pitch motions of a spar platform; surge and heave motions of risers, and vertical motions of water inside the moon pool. The results indicated that the heave amplitude of the platform is reduced significantly by the coupled motions of the platform and water inside the moon pool.

The study presented in this paper investigated the heave motions of a truss spar platform considering the dynamic coupling between the platform motions and the motions of water in the moon pool. The effect of the motions of water on the heave motions of the platform was investigated. The platform heave and the moon pool motions were discussed as opening ratios of the bottom guide plate varied. Three detachable guide plates were designed for the model tests, w i t h 0% opening ratio (moon pool totally closed, Case 1), 30% opening ratio (Case 2) and 70% opening ratio (Case 3). The results of the experiments were used to validate the numerical analysis.

2. Dynamical modeling

The vertical motion equations of water inside the moon pool of a spar platform were derived based on the work of Gupta et al. (2008) through the following steps: (1) the conservation laws of mass and momentum of water inside the moon pool were applied to find the pressure acting on the top of the guide plate; (2) an empirical formula to calculate the pressure acting on the bottom of the guide plate was used; (3) Newton's second law was applied to the fluid i n the guide plate gap, to let the pressure difference between t w o sides of the guide plate accelerate the fluid. The heave equations of the spar platform was established by applying the rigid body dynamics equations given by Liu et al. (2014), and the wave forces were calculated based on the potential theory.

The coordinate system and cross section of the platform hull are shown i n Fig. 1. In Fig. 1(a), x axis is i n the horizontal plane of the still water, z axis is vertically upward and through the gravity center of the platform, - Z o is the hull static draft, Z i is the moon pool water surface elevation i n the moon pool relative to the horizontal plane of the still water In Fig. 1(b), Sg is the open area at the bottom of the moon pool, So is the area of the guide plate and Sp is the sectional area of the hard tank. The moon pool sectional area can be written as Smp=zSo+Sg and the moon pool opening ratio can be written as

Sg/Smp-First, the motion equation of water i n the moon pool in vertical direction is derived. Taking the moon pool water as a deformable control volume, the mass conservation equation is

p C v i V t ) X ndy4=

-i ,

where p is the water density, V i is the velocity of water flowing through the guide plate gap, the magnitude is Vg. 'Vb is the velocity of the spar platform heave, the magnitude is z, n is unit outward normal vector, Mmp = pSmpizi -Z-ZQ) is the mass of the moon pool water. Substituting above parameters into Eq. (1) leads to

Z + Sn Sg X (Z, - Z ) (2) still water hard tank O moonpool

bottom guide plate

Fig. 1. Coordinate system and cross section of the hull of the platform: (a) coordinate system, (b) cross section.

The linear momentum equation of the deformable control volume can be w r i t t e n as Hansen (1967)

(3)

where V is fluid velocity vector i n the control volume CV, Vr is the fluid velocity vector cross the control surface CS.

To simplify the analysis, assuming the flow properties over the cross section and those inside the control volume are uniform, respectively (White, 2003), the right hand side of Eq. (3) can be further w r i t t e n as

-Vb)

ndA = ^{pZiSmp(Zi - z - z o ) } - p z i (vg-z)Sg (4)

where ^2 is the velocity of the water particle inside the moon pool, the magnitude is Z i .

The left hand side of Eq. (3) includes surface force acting on the control surface and mass force acting on the water particles inside the moon pool. The forces can be w r i t t e n as

= -pgSmp(Zi-Z-Zo) + p-iSmp (5)

where is the pressure on the upper side of the guide plate, g is the acceleration of gravity.

Substituting Eqs. (4) and (5) into Eq. (3) yields the relation

pZ-iSmp(Z^ - Z - Z o ) = -pgSmp(Zi-Z-Zo)+PiS,

From Eq. (6), can be derived as

Pl=PiZx+g)(Zi-Z-Zo)

(6)

(7)

The pressure on the lower side of the guide plate includes the static pressure, the inertia force due to vertical acceleration, the dynamic pressure due to velocity and the incident wave pressure.

164 I , Uu et aL / Ocean Engineering 92 (2014) 162-174 It can be given by a semi-empirical formula (Gupta et al., 2008)

P2 = -PS(Z+Z0) - aipdgZ+Ypdgüw +Pw (8)

where y is the wave acceleration, a,v is the wave coefficient i n the moon pool, dg is the equivalent diameter of the moon pool opening, 's the wave pressure acting on the guide plate, as is the coupling mass coefflcient of platform to moon pool.

The pressure difference between t w o sides of the guide plate accelerates the water i n the guide plate gap. Applying the force balance principle to the water in the guide plate gap, the following equation can be obtained

( P 2 - P i ) S g = MgVg4-^p(v„ - Vg+i:)|v,v - Vg +z\[<gSg ₍₉₎

where Mg=pa-iSgdg, «i is the added mass coefficient of the moon pool, v,v is the wave velocity, Kg is the damping coefflcient of the water flowing through the guide plate gap.

Substituting Eqs, (7) and (8) into Eq. (9), the motion equation of water in the moon pool can be obtained as follows

Zl [paidgSmp + {z-i -z-Zo)pSg] +z^pgSg = z[pai(Smp-Sg)dg-pa3dgSg] + [v,v-5mp(Zl - Z ) / S g ] |Vw -Smp(Zi -Z)/Sg\-pI<gSg+p„Sg+Ypa„dgSg '

(10) Second, the heave motion equation of the spar platform is established. Considering the inertia force, restoring force, damping force, wave force and force f r o m the moon pool water, the heave motion equation of the spar platform can be written as

Z(M + i^M)+PgSmZ + B^Z + B2Z\Z\=p^Sp + Fm (11)

where M is the platform mass, A M is the heave added mass, Sm is the water plane area of the platform hull, Bi is the linear damping coefficient, B2 is the quadratic damping coefficient, is the force acting on the guide plate. The risers and the mooring system are not considered here.

Weggel and Roesset (1994) derived the expression of total vertical diffraction force acting on truncated cylinder. Based on their work, the wave pressure acting on the Spar platform in heave direction can be represented as

Pw=PSH^^ 1 - 1 sin(feR) ' ^ - l f > ) e - " c o s ( . t - . ) (12)

where Hw is the wave height, R is the hull radius, k is the wave number, is the first kind Bessel function, co is the wave frequency, e = 31 nikR^ • V 1 8 0 .

The force acting on the guide plate can be divided into t w o parts as

fm = F p + f c (13)

The flrst term in the right hand side of Eq. (13) is the pressure force, it can be represented as

Fp = (P2-P0(Smp-Sg) (14)

To simplify the calculation, only the static parts of P2 are considered. Substituting Eqs. (7) and (8) into Eq. (14) leads to

Fp = { P w - Z i P g - Z i ( Z i -z)p+z-ipZQ}iSmp- -Sg) (15)

The second term on the right hand side of Eq. (13) is a correction term, i t accounts for the effect of the net fluid accel-eration on the pressure acting on the platform's bottom. It can be represented by a semi-empirical formula (Gupta et al., 2008), as follows

where 04 is the coupling mass coefficient of moon pool to platform.

Substituting Eqs. (13), (15) and (16) into Eq. (11), the heave motion equation of the platform can be expressed as

z[M+AM-paidg(Sp +Smp - Sg)]+ZpgSm +B^Z+B2Z\Z\ = p„(Sp+Smp - Sg)

-Z,pg(Smp-Sj)-Z](Z,-Z>(S,„p-Sg)-|-Z,|>(Smp-Sg)Zo-^a4df(Sp+Smp-Sg)] (17) Eqs. (10) and (17) can be combined to form the 2-DOF coupling motion equations of the heave motions of the Spar platform and the vertical motions of the moon pool water. The coupling terms include the inertia force, the damping force, the restoring force and the added mass came f r o m the mass coupling between the platform and the moon pool.

3. Platform parameters

A hypothetical spar platform geometry was used to carry out the analysis, w h i c h is similar to Horn Mountain platform (Fetter, 2000). The platform parameters are shown in Table 1.

The risers of Horn Mountain Spar platform are supported by a buoyancy tanks. For the present hypothetical platform, the top tension risers are used. The objective is to investigate the effect of water motions in the moon pool on the heave motions of the spar platform. Three cases are considered as listed i n Table 1, including the moon pool is totally closed (Case 1), 30% of the moon pool is open (Case 2) and 70% of the moon pool is open (Case 3). The draft and the CG (center of gravity) of the platform are equal for the three cases. The natural heave period of the platform for Case 1 is ts = 24.4 s, for Case 2 and Case 3 is ts = 26.8 s. Using the empirical relation tm = 2K^/d/g, where d is the draft of hull, the natural period of the moon pool motions i n the vertical direction is

tm = 14.7 s.

4. Numerical simulation results and discussion

The heave responses of the platform and vertical motions of water in the moon pool are predicted in this section. For the truss spar platform, the radiation damping is insignificant relative to the viscous damping (Lu et al., 2003; Ghosh and Spanos, 2009), therefore it is neglected and only the nonlinear damping is considered in the following calculations. The nonlinear damping coefficients for the three cases are obtained f r o m the model experiments as detailed in Section 6. The initial values of the coupling parameters and the moon pool hydrodynamic coeffi-cients are a\ = 0.326, « 3 = « 4 = 0.5, ay, = 1.45 and Kg = 1.0. Their effects on the responses w i l l be analyzed i n Section 5.

Solving Eqs. (10) and (17) by Runge-Kutta method, the time histories of the platform heave responses z and the vertical Table t Platform parameters. Fc = (Z-Zi)/Jo;4dg(Sp +Smp - S g ) (16) Description Value Water depth 1652 m Total length 169.16 m

Diameter of hard tank 32.31 m

Draft 153.92 m

Center of gravity from keel 90.39 m

Moon pool 15.85 X 15.85 m^

Length of hard tank 68.88 m

Number of heave plates 3

Opening ratio/Case 1 0% (Moon pool totally closed)

Opening ratio/Case 2 30%

L. Liu et al. / Ocean Engineering 92 (2014) 162-174 165 800 BOO BSO 700 time (s) -IOO 500 550 600 time (s) 650 700 750 BOO 550 600 time (s)

Fig. 2. Response time histories for Case 1, wave height 4 m: (a) wave period 10 s, (b) wave period 20 s, (c) wave period 28 s.

Q -4 \

500 600 650 700

time (S)

800

Fig. 3. Response time histories for Case 2, wave height 4 m: (a) wave period 10 s, (b) wave period 20 s, (c) wave period 28 s.

motions of water i n tlie moon Z i are obtained, as shown i n Figs. 2-4.

It is observed that for Case 2 and Case 3, the amplitudes of z and

Z l depend on the wave period. When the wave period is close to tm

(the natural period of water motions in vertical direction i n the moon pool), larger vertical motions of the moon pool are excited. When the wave period is close to ts (the natural period of the platform heave), larger heave motions of the platform are excited. For the shorter wave periods, the platform heave and the moon pool motions are nonlinear, as shown i n Fig. 2(a) and Fig. 4(a).

In order to consider the nonlinear damping sufficiently, ampli-tude response curves of the platform heave and the moon pool motions are calculated using a numerical iterative method (Li and Ou, 2009) through the following steps: (1) the mean response amplitudes of the platform heave and the moon pool motions are calculated when hw = 4 m as wave periods vary; (2) each response amplitude is divided by the wave amplitude. The results for above three cases are shown i n Fig. 5.

In Fig. 5, rUa and indicate amplitudes of the moon pool motions and the heave motions of the platform, respectively, Wa indicates the wave amplitude. It is observed that the motions of the moon pool greatly depend on the opening ratio of the moon

pool. Comparing w i t h Case 2, the maximum amplitude of the moon pool motions i n Case 3 is nearly 2.4 times of those in Case 2 and large amplitude motions of the moon pool are excited. The maximum heave amplitudes of the platform i n Case 2 and Case 3 are smaller than those of Case 1, and the platform heave motions decrease for the semi-closed cases comparing w i t h the totally closed case. The maximum values of p j w a i n Case 1, Case 2 and Case 3 are 2.33,1.92 and 2.08, respectively, as shown in Fig. 5(b). Comparing w i t h Case 1, it reduces nearly 17.6% i n Case 2 and reduces nearly 10.7% in Case 3,

For Case 3, heave response amplitude curve of the platform has two peaks, one is close to t j and the other one is close to tm- When the wave period is close to tm, large heave motions of the platform are excited due to large amplitude motion of the moon pool and the energy is transferred to the platform due to the dynamic coupling.

5. Parametric analysis

In this section, the amplitude response curves as stated in Section 4 are calculated for the cases of 30% and 70% opening ratios to investigate the effects of coupling parameters.

166 L Liu et al. / Ocean Engineering 92 (2014) 162-174 0,4 0,0 Q .0.2 -0,4 500 600 650 700 time (s) 760 300

Fig. 4. Response time histories for Case 3, wave height 4 m: (a) wave period 10 s, (b) wave period 20 s, (c) wave period 28 s.

5.J. Coupling mass coefficients

The influences of the coupling mass coefficients as and «4 are investigated by calculating the response amplitude curves for the different coefficient values, as shown i n Figs. 6-9.

It can be seen that 03 greatly affects the vertical motions of the water inside the moon pool, the amplitude and the natural period of the moon pool motions increase w i t h the increase of « 3 , since « 3 affects the inertia force of moon pool motions. In Case 3, when the wave period is close to t„„ « 3 obviously affects the platform heave motions because of strong dynamic coupling between the moon pool and the platform heave, as shown i n Fig. 7(a). « 4 affects the platform heave and i t increases w i t h the increase of « 4 . Especially i n Case 3, the smaller peak of the heave response amplitude curve of the platform becomes more obvious w i t h the increase of « 4 , as shown i n Fig. 9(a).

5.2. Hydrodynamic coefficients of water in the moon pool

This section discusses the influences of hydrodynamic coeffi-cients of the water i n the moon pool, including a i , Kg and ow, the results are shown i n Figs. 10-15.

I

12 14 16 16 20 22 24 26 28 30

wave period (s)

14 16 13 20 22 24 26 28 30 32 wave period (s)

Fig. 5. Response amplitude operator curves: (a) the moon pool motions, (b) the platform motions. 1.0 0,5 0,0 — — « 3 = 0 - 2 — ^ « 3 = 0 . 6 - ^ ^ « 3 = 1 . 0 12 14 16 IS 20 22 24 2B 28 30 32 wave period (s)

I

Fig. 6. Effect of «3 on platform and moon pool motions. Case 2: (a) platform heave motions, (b) vertical moon pool motions.

It can be found that a\ affects the inertia force of moon pool motions and therefore greatly affects the amplitude and the natural period of the moon pool motions, as shown in Fig. 10(b)

168 L. Liu et al. / Ocean Engineeiing 92 (2014) 162-174

Fig. 12. Effect of Kg on platform and moon pool motions, Case 2: (a) platform Fig. 14. Effect of a„ on platform and moon pool motions. Case 2: (a) platform heave motions, (b) vertical moon pool motions. heave motions, (b) vertical moon pool motions.

L Uu et al. / Ocean Engineering 92 (2014) 162-174 169

O Q L . U L . 1 1 1 _ L , 1 1 1

12 14 16 18 20 22 24 26 28 30 wave period (s)

Fig. 15. Effect of aw on platform and moon pool motions, Case 3: (a) platform lieave motions, (b) vertical moon pool motions.

a 4 r

I I -> 1 1 < 1 , , 1 r

12 14 16 18 20 22 24 26 23 30 wave period (s)

Fig. 16. Effect of nonlinear damping coefficients. Case 2: (a) platform heave motions, (b) vertical moon pool motions.

and Fig. 11(b). The motions of the moon pool are sensitive to Kg and i t decreases w i t h the increase of Kg, for example, in Fig. 13(b) when Kg increases f r o m 0.8 to 1.5, the maximum value of rria/Wa

decreases nearly 25%.

For Case 3, ai and Kg affect amplitude of the heave motions of the platform when wave period is close to tm, as Fig. 11(a) and Fig. 13(a) showing. The reason is that the motions of the moon pool are greatly affected by «i and Kg, the energy is transferred to the platform heave due to strong dynamic coupling when the opening ratio is large. Coefficient aw has little effect on the platform heave and the moon pool motions, as shown in Figs. 14 and 15.

11 I I i 1 1 1 1 1 1

12 14 16 18 20 22 24 28 28 30 wave period (s)

Fig. 17. Effect of nonlinear damping coefficients. Case 3: (a) platform heave motions, (b) vertical moon pool motions.

0 I 1 - 1 1 1 i - i t 1 1 I 1 '

0.0 0,2 0.4 0,6 0.8 1,0 1,2 1.4 1,5 1.6 2,0 2.2

170 L Liu et al. / Ocean Engineering 92 (2014) 162-174

5.3. Damping coefficients

This section discusses the influence of damping coefficients. The initial damping coefficient B2 is obtained f r o m model experi-ments, as presented i n Table 3 of Section 6. The response amplitude curves for different nonlinear damping coefficients O.2B2, O.5B2, O.8B2, I.5B2 and 2.OB2 are calculated and the results are shown i n Figs. 16 and 17, The maximum values of heave response amplitude curves of the platform for different damping coefficients are shown i n Fig. 18.

Fig. 16(a) and Fig. 17(a) show that the heave motions of the platform decreases w i t h the increase of nonlinear damping. The response amplitude curve of the moon pool motions has two peaks for lower nonlinear damping, one is near to the natural period of the moon pool the other is near to the natural heave period of the platform, as shown in Fig. 16(b) and Fig. 17(b). This indicates that the effect of the platform heave on the moon pool water motions becomes important for the lower nonlinear damp-ing, especially i n the case of smaller moon pool opening ratio.

It is observed f r o m Fig. 18 that the maximum values of the heave response amplitude curves of the platform nonlinearly decrease w i t h the increase of nonlinear damping. When the nonlinear damping coefficient increases f r o m O.2B2 to O.5B2 the platform heave decreases sharply. Comparing Case 2 w i t h Case 3, the heave motions of the platform are greatly affected by the nonlinear damping i n Case 3.

6. Model experiments

The model experiments were used to study the effect of the moon pool water on the heave motions of the truss Spar platform. The experiments were carried out in Tianjin University's wave tank w i t h size of 137 m long, 7 m wide, and 3.5 m deep and is equipped w i t h a flap type wave maker. The wave lengths made by the wave maker are f r o m 2 m to 12 m.

6.J. The model and test conditions

The model scale 1:130 was selected taking into consideration the size of the tank and the wave making capacity of the wave maker. Model parameters are shown in Table 2.

Three detachable bottom guide plates w i t h different opening ratios were constructed and three test cases were carried out including 0% opening ratio (moon pool was totally closed, Casel), 30% opening ratio (Case 2) and 70% opening ratio (Case 3) as illustrated i n Fig. 19, which shows the photographs of guide plates installed on the platform (photographed f r o m top to bottom of the moon pool). Drafts of the platform were kept constant and the centre of gravities were almost equal for the three cases.

The motions o f t h e model were measured in regular waves. The heave natural periods of the platform for three cases are f r o m 24 s to 27 s and the natural period of vertical motion of water inside

Table 2 Model parameters.

Parameter Unit Value

Total length m 1.300

Draft m 1.185

Total displacement t 0.021

Diameter of hard tank m 0.249

Length of hard tank m 0.530

Heave plates space m 0.176

Cross section area of heave plate m^ 0.249 X 0.249 Cross section area of moon pool m^ 0.120x0.120

a

Fig. 19. Three different bottom guide plates: (a) Case 1, (b) Case 2, (c) Case 3. the moon pool is 14.7 s. So, i n the experiment the wave periods were changed f r o m 12 s to 32 s and the wave height 4 m was selected.

L Liu et al. / Ocean Engineering 92 (2014) 162-174 171

Fig. 20. Sketch of the test setup.

Table 3

Natural period of the platform heave and nonlinear damping coefficient. Heave natural period (s) Damping coefficient

Case 1 24.40 0.0797

Case 2 26.79 0.1211

Case 3 26.79 0.1126

Fig. 21. The test layout.

6.2. The test setup

A wave probe and an optical motions measurement system were used to measure the incident wave displacement and 6-DOF motions of the model as illustrated i n Fig. 20, which shows the sketch of test setup. The model was moored by four springs attached at the center of pitch to l i m i t the horizontal motions, three LEDs were mounted on the top of the model to mark the motions of the model as illustrated in Fig. 21. The camera was mounted i n a steel frame fixed i n towing tank. The distance between the camera to the model is about 1.5 m.

Before the test, the wave probe and the optical motion measuring instrument were calibrated. For all the tests, the camera was kept fixed in order to minimize the errors in measured results. Each test can't be carried out until the water surface becomes calm. Data were collected before the front of the first wave reaches the wave probe and were stopped as soon as the wave reaches the beach at the far end of the towing tank. Only the harmonic part of the measured signal were used in the following motions analyses. The measured results can be applied to validate the numerical solution. In following Sections 6.3 and 6.4 all the data presented were i n the prototype scale.

6.3. Spar platform motions in calm water

The heave free decay motions of the spar platform in calm water were measured for the three cases and the damping coefficients were obtained.

The motion equation can be written as

x+diX+d2X\x\+d3X = 0 (18)

where d j is the linear damping coefficient and d2 is the quadratic damping coefficient, di and d2 can be determined by the free decay motions using the method proposed by Faltinsen (1990). Only the nonlinear damping coefflcient d2 was considered here.

The heave natural period was obtained by average of 10 consecu-tive period cycles of the free decay test. The results for the three cases are shown in Table 3.

Table 3 shows that natural period of the spar platform i n heave in Case 2 and Case 3 are longer than that in Case 1. Comparing w i t h Case 1, d2 increases about 52% in Case 2 and increases about 41% i n Case 3. The nonlinear damping of the platform in heave is much more enhanced for the cases of semi-closed moon pool. Comparing Case 2 w i t h Case 3, d2 decreases a little i n Case 3.

6.4. Spar platform motion in waves

The 6-DOF motions of the Spar platform were tested i n different waves and the platform heave motions were analyzed. Some response curves of the platform heave are shown in Figs. 22-24, the amplitude response curves for the three cases are shown in Fig. 25.

It can be found that the heave amplitude of the platform increases w i t h the wave period approaching natural period of the platform i n heave, and the heave responses are nonlinear for the shorter wave periods, as shown i n Fig. 22(a), Fig. 23(a) and Fig. 24(a). These conclusions agree w i t h the numerical results of Figs. 2-4.

To make a comparison between the results of numerical calculation and those of model test, the results of numerical calculation were also presented i n Fig. 25(a). It is shown that the maximum heave amplitude of the platform in Case 1 is bigger than those i n Case 2 and Case 3. Comparing w i t h Case 1, it decreases nearly 20% in Case 2 and decreases nearly 13% i n Case 3. This agrees well w i t h the numerical results and approves validity of the dynamic predictions.

There are two peaks i n the heave response amplitude curve of the platform for Case 3, the higher one is near to the heave natural period of the platform and the lower one is near natural period of motions of the water inside the moon pool as shown in Fig. 25(b). The results agree w i t h the numerical results of Section 4. The maximum value of the lower peak of the tests is smaller than that of the numerical calculations. Since i t is difficult to exactly match the wave period generated by the wave maker to the natural period of water motions i n vertical direction inside the moon pool in the model experiment. The dynamical equations only consid-ered the heave motions of the platform, while i n the test the

172 L Liu et al. / Ocean Engineering 92 (2014) 162-174 800 900 time (s) 1100 800 900 time(s) 1100 1100 600 7Ü0 800 900 time(s) 1000 1100 a> 800 300 tirne (s) 110D

Fig. 22. Platform lieave time liistories for Case 1, wave heigtit 4 m: (a) wave period 16.5 s, (b) wave period 22.0 s, (c) wave period 26.2 s.

model has 6-DOF motions. This leads to the difference between the RAO curves obtained by the model test and those obtained by the numerical calculation.

7. Conclusions

The research reported in this paper investigated the heave response of a truss Spar platform w i t h semi-closed moon pool. The 2-DOF coupled dynamic motion equations of the platform i n heave and the vertical water motions i n the moon pool were derived. The calculations and experimental measurements were carried out for three cases where the moon pool was totally closed, 30% and 70% open. The effect of the moon pool motions on the heave motions of the platform was analyzed. The heave characteristics of the plat-form and the vertical motions of the moon pool were discussed for different parameters. IVIodel experiments were carried out to

1000

Fig. 23. Platform heave time histories for Case 2, wave height 4 m: (a) wave period 15.8 s, (b) wave period 23.9 s, (c) wave period 28.5 s.

validate the numerical calculations. The main conclusions are listed as follows:

(1) Comparing the case when the moon pool was totally closed w i t h the cases when the moon pool was partially closed, the heave motions of the platform decrease in the cases of semi-dosed moon pool. Different moon pool opening ratios result in different amplitudes of the heave motions of the platform. In the actual design, an optimized opening ratio can be deter-mined to decrease heave of the spar platform. In the present study the heave motions of the platform are lower i n case of 30% opening ratio.

(2) A lower peak appears i n the heave response amplitude curves of the platform i n Case 3 when wave period is near to the natural period of the moon pool motions (tm). The reason is that when the wave period is close to t,,,, larger moon pool motion amplitudes are excited and energy is transferred to the

L Liu et al. / Ocean Engineering 92 (2014) 162-174 173 800 900 tirne (s) 1100 10 12 U IB 18 20 22 24 26 28 30 32 wave period (s) 0) 020 -0,5 6 0 0 650 7 0 0 760 8 0 0 850 9 0 0 9 S 0 1000 1050 1100 t i m e (s) 800 850 900 time (s) 950 1000 1050 1100

Fig. 24. Platform heave time histories for Case 3, wave height 4 m: (a) wave 15.5 s, (b) wave period 22.0 s, (c) wave period 28.5 s.

period

platform due to strong dynamic coupling. Thus coupling of the moon pool water should be considered in the cases of semi-closed moon pool.

(3) The motions of the moon pool are mainly affected by the coupling mass coefficient « 3 , hydrodynamic coefficients ai and Kg. The influence w i l l be transferred to the heave motions of the platform in cases of larger opening ratio of the moon pool due to the dynamic coupling. The platform heave is mainly affected by the coupling mass coefficient « 4 . Hydro-dynamic coefficient aw has little effect on the heave motions of the platform and the motions of the moon pool.

(4) The nonlinear damping is much more enhanced in the cases of semi-dosed moon pool. The maximum value of amplitude response curve of the platform heave nonlinearly decreases w i t h the increase of nonlinear damping.

In this paper the nonlinear damping coefficient of platform heave was obtained f r o m the model experiments. It can be

0,15

I

Fig. 25. Response amplitude curves of the platform heave by the experiment; (a) the whole graph, (b) the local magnification graph.

calculated by the CFD (computational fluid dynamic) method, this is our research work of next step.

Acknowledgments

The project was supported by the Natural Science Foundation of China under Grant no. 51179125, the Innovation Foundation of Tianjin University under Approving no. 1301 and also supported by China Scholarship Coundl (CSC).

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